Final answer:
The initial mass of the sample was 40 mg. The mass 8 weeks after the start is 0.625 mg.
Step-by-step explanation:
The half-life of Palladium-100 is 4 days. This means that every 4 days, half of the Palladium-100 in the sample will decay. After 12 days, we can calculate how many half-lives have passed by dividing the total time by the half-life: 12 days / 4 days = 3 half-lives. Since half of the sample decays with each half-life, the mass remaining after three half-lives is calculated by multiplying the initial mass by (1/2) raised to the power of the number of half-lives: 5 mg = Initial mass * (1/2)^(3). We can solve this equation to find the initial mass: Initial mass = 5 mg / (1/2)^(3) = 5 mg / (1/8) = 40 mg.
To find the mass 8 weeks after the start, we need to convert 8 weeks to days: 8 weeks * 7 days/week = 56 days. We can now calculate the number of half-lives that have passed: 56 days / 4 days = 14 half-lives. Using the same formula as before, we can find the mass after 14 half-lives: Mass after 14 half-lives = Initial mass * (1/2)^(14) = 40 mg * (1/2)^(14) = 0.625 mg.